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Function g is defined as g(x)=f(2x). What is the graph of g?

The problem involves analyzing the graph of a function \( f(x) \) and determining the graph of a new function \( g(x) = f(2x) \).

### Understanding the Problem:
The function \( g(x) = f(2x) \) represents a horizontal compression of the function \( f(x) \). Specifically, every point on the graph of \( f(x) \) is moved closer to the y-axis by a factor of 2.

### Step-by-Step Explanation:
1. **Horizontal Compression**: The transformation \( f(2x) \) compresses the graph of \( f(x) \) horizontally by a factor of 2. This means that for any point \((x, y)\) on the graph of \( f(x) \), there is a corresponding point \((\frac{x}{2}, y)\) on the graph of \( g(x) \).

2. **Applying to the Graph**:
   - The vertex of the parabola \( f(x) \), originally at \((4, -4)\), will now be at \(\left(\frac{4}{2}, -4\right) = (2, -4)\) for \( g(x) \).
   - The point \( (2, 0) \) on the graph of \( f(x) \) will now be at \( (1, 0) \) on the graph of \( g(x) \).
   - Similarly, the point \( (6, 0) \) on the graph of \( f(x) \) will be at \( (3, 0) \) on the graph of \( g(x) \).

3. **Graph of \( g(x) \)**:
   - The graph of \( g(x) \) will still be a parabola, but it will be narrower than the graph of \( f(x) \).
   - The x-intercepts of \( g(x) \) will be at \( x = 1 \) and \( x = 3 \), and the vertex will be at \( (2, -4) \).

### Summary:
The graph of \( g(x) = f(2x) \) will be a horizontally compressed version of the graph of \( f(x) \) with the vertex at \( (2, -4) \) and x-intercepts at \( x = 1 \) and \( x = 3 \).

Would you like to see a visual representation of the graph of \( g(x) \)? Do you have any questions or would you like further details?

Here are some related questions you might consider:

1. How would the graph change if \( g(x) \) were defined as \( g(x) = f\left(\frac{x}{2}\right) \)?
2. What happens to the graph if \( g(x) = 2f(2x) \)?
3. How do vertical shifts affect the graph of a function?
4. What does the transformation \( g(x) = f(x - 2) \) do to the graph of \( f(x) \)?
5. How can you determine the new x-intercepts after a horizontal compression?

**Tip:** Remember, horizontal transformations like \( f(2x) \) affect the x-values of the function, compressing or stretching the graph horizontally.

Explore the graph of g(x) = f(2x), a horizontally compressed version of the function f(x). Learn how every point on the graph of f(x) moves closer to the y-axis by a factor of 2, affecting key features like the vertex and x-intercepts. Suitable for high school students studying function transformations.

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